Resonance Frequency Calculator

What Is the Resonance Frequency Calculator?

How to use the resonance frequency calculator

1. Choose the unknown. The Solve for menu opens on Natural frequency, and the other two choices are Spring constant and Mass. Any two of the three fix the third.
2. Type the spring constant. It takes N/m or kN/m and opens on 200 N/m. This page asks you for the stiffness rather than deriving it from a material or a coil geometry.
3. Type the mass. It takes kg or g and opens on 2 kg. Count only the part that actually moves, not the frame it is mounted on.
4. Type the damping coefficient. It is in N s/m and opens on 2 N s/m. It is an input in every mode, so it never leaves the form — and it never touches the headline.
5. Type the drive frequency. It takes Hz or kHz and opens on 1.592 Hz. This is the frequency something outside is pushing at, and it is an input in every mode too.
6. Read the answer. The headline always carries four significant figures, with trailing zeros dropped, in the unit that box was showing — switching to scientific notation outside the range 0.0001 to 10,000,000. Set the unit while the quantity is still an input and then point Solve for back at it.
7. Read the damping ratio and Q first. At the defaults they read 0.05000 and 10.00, and neither moves when you change the drive. Q is how many times the static deflection the swing reaches when the drive sits on the natural frequency.
8. Read the ratio, the magnification and the phase lag. These three describe your drive, not the oscillator. A phase lag of exactly 90.00 degrees means the drive is exactly on the natural frequency, whatever the damping.
9. Read the peak response. It is always a little below the natural frequency, and past a damping ratio of about 0.7071 it prints the word none because there is no peak left to report.
10. Open Show working. The steps rearrange the relation, then give the damping ratio, Q, the drive ratio, the magnification, the phase lag, the peak frequency, the approximate half-power width and your figures in every unit the boxes offer.

This page exists for the one step that turns a spring and a mass into a frequency, and then says how big the response at that frequency gets. The resonance in physics guide is the prose half of it, with the mechanism, the worked problems and the misconceptions set out in full. What the mass is doing once it is moving — the sinusoid, the peak speed, the peak acceleration — is simple harmonic motion, a separate story told on its own page, and the simple harmonic motion calculator is the tool for it.

Three mistakes account for most wrong answers here. The first is expecting the damping box to move the headline: it cannot, because neither c nor the drive frequency appears in f0 = (1/2pi) sqrt(k/m). The second is reading the magnification as a distance — it is a multiplier, and you need your own static deflection to turn it into millimetres. The third is treating the peak response reading as “the resonance frequency”: the peak moves with the damping while the 90.00 degrees phase lag does not.

The spring constant is an input here and the page never measures it. If you do not have a figure for it, the spring constant calculator works it out from a force and an extension, and the amplitude guide is where the height of a vibration belongs as a subject in its own right — on this page the amplitude is only ever an output, and a dimensionless one.

Worked example: change one thing at a time

The table starts at the defaults, moves the unknown through all three quantities, then loads each preset in turn, walks every unit menu, and finally reaches the entries the calculator declines. Rows 1 to 3 are one set of boxes solved three ways; rows 4 to 6 are the same thing once the drive carries eight figures instead of four. Every answer and reading cell was read out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool.

Rows 1 to 3 are the rounding trap, caught in the act. The boxes open on 1.592 Hz, 200 N/m and 2 kg, and those three cannot all be exactly true at once because pi is in the relation: 200 N/m on 2 kg is 1.5915494 Hz, and the four-figure 1.592 Hz is 0.02831 per cent above it. Stiffness goes as the square of the frequency, so that doubles, and row 2 returns 200.1 N/m rather than 200. Row 3 goes the other way to 1.999 kg. Nothing is wrong: a frequency typed to four figures simply does not pin a stiffness to four figures.

Rows 4 to 6 remove it. The first preset writes the natural frequency and the drive as 1.5915494 Hz, eight significant figures, and all three directions then agree — 1.592 Hz, 200 N/m, 2 kg — with the magnification at a clean 10.00 and the phase lag at 90.00 degrees. That is the state the whole cluster quotes, and it is one click away from the one the page opens in.

Rows 7 to 10 change the damping and nothing else. The headline does not move from 1.592 Hz in any of them, which is the first claim of the page in four rows. What does move is the magnification: 50.00, 1.000, 0.7143, 0.7042, a range of 71 from the same spring, the same mass and the same push. Rows 9 and 10 differ by 0.4 N s/m and sit either side of the threshold: at 0.7000 there is still formally a peak, at 0.2251 Hz, but its magnification is 1.000 — the same as the static response, so there is nothing a reader could see. At 0.7100 the reading becomes the word none.

Rows 11 and 12 move the drive alone. At half the natural frequency the magnification is 1.330 and the phase lag 3.814 degrees: the mass follows the push almost in step and barely amplifies it. At twice the natural frequency the magnification is 0.3326 and the lag 176.2 degrees: the mass moves the opposite way to the push and less than a steady push would move it. Between those two sits the factor of 10 at resonance.

Rows 13 and 14 are the two scalings. Four times the stiffness doubles the frequency to 3.183 Hz; four times the mass halves it to 0.7958 Hz. Both leave the damping coefficient at 2 N s/m, yet the damping ratio halves to 0.02500 in each case and Q doubles to 20.00 — because the ratio is c divided by 2 sqrt(mk), and both changes raise sqrt(mk).

Rows 15 to 17 are the two back-solves and a retune. A 0.5066 kg mass needs a 45 N/m spring to sit at 1.500 Hz; a 157.9 N/m spring needs a 0.25 kg mass to sit at 4.000 Hz. Row 17 is the practical one: to pull the default oscillator off 1.592 Hz down to 1.000 Hz on the same spring takes 5.066 kg, so you add 3.066 kg — and the frequency ratio 1.592 is exactly the square root of 5.066 over 2.

Rows 18 to 21 change nothing but the units. 2000 g is the same 2 kg, 0.2 kN/m the same 200 N/m and 0.001592 kHz the same 1.592 Hz, and all three return the identical 1.592 Hz with identical readings. Row 21 shows the other half of the rule: the answer comes back in whichever unit the target box was showing, so the spring reads 0.2001 kN/m there rather than 200.1 N/m.

Rows 22 to 30 sit outside the domain and are declined there rather than answered. A zero mass makes the frequency infinite and a zero spring makes it zero; a negative mass or spring puts a negative number under the square root. A zero damping makes Q and the magnification at resonance infinite, which is the one place this subject invites a wrong answer, so it is refused. A zero drive frequency has no ratio to the natural one, and a zero or negative natural frequency cannot be squared into a stiffness. In every case the widget says so in words instead of printing an infinity or a NaN.

Formula and symbol reference

One relation does the first half of the work. A spring pulls back in proportion to how far it is stretched and a mass resists being accelerated, and the balance between them gives an angular frequency of sqrt(k/m). Divide by 2π to get a frequency in hertz: f0 = (1/2pi) sqrt(k/m). Rearranged the other two ways that is k = m(2πf0)2 and m = k / (2πf0)2, and those are the three positions of the Solve for menu.

The second half is what happens when something drives the mass. Two dimensionless numbers settle it: the damping ratio ζ = c / (2 sqrt(mk)), and the frequency ratio r = f / f0. The magnification is then M = 1 / sqrt((1 - r2)2 + (2ζr)2) and the phase lag is atan2(2ζr, 1 - r2). Put r = 1 into the first and everything but the damping term cancels, leaving M = 1 / (2ζ), which is Q exactly.

The spring constant is the k of the force-extension relation, and the Hooke's law calculator is where that relation itself lives. For a reader who arrives with a period rather than a frequency, the frequency formula guide is the conversion — the period of the default oscillator is 0.6283 s. And the same shape of formula with different ingredients describes a swinging pendulum, where the length and gravity replace the spring and the mass: the pendulum period calculator is that tool, and no figure crosses between the two pages.

One oscillator, one resonance — and that is a restriction worth naming. A stretched string has a whole ladder of resonances rather than one, which is why the standing waves and harmonics guide reads so differently from this page; the standing waves simulator draws that ladder. A tuned electrical circuit obeys the same mathematics again with an inductance and a capacitance in place of the mass and the spring. Neither has a box here and neither is given a figure.

The physics: why the damping decides how big resonance gets

Push the mass at the wrong frequency and you spend half of every cycle fighting the motion you created in the other half. Push it at its own frequency and every push arrives while the mass is already moving the way you are pushing, so the work you do accumulates instead of cancelling. The swing then grows until the energy the damping removes each cycle equals the energy you put in — and that balance, not the size of the push, is what fixes the final amplitude.

That is why the magnification at the natural frequency comes out as exactly Q = 1 / (2ζ) and contains no force at all. Halve the damping and the swing doubles. Load the light damping and heavy damping presets one after the other: the headline stays at 1.592 Hz byte for byte while the magnification goes from 50.00 to 1.000, a factor of 50 from the same spring, the same mass and the same push.

Three things on this page respond to three different inputs, and the readings are arranged so you can see it one change at a time. Move the spring or the mass and the headline moves and so does the damping ratio. Move the damping and the headline does not move at all while the damping ratio, Q and the magnification do. Move the drive and only the last four readings move. The free, undriven version of this oscillator is worth watching alongside: the simple harmonic motion simulator shows the same mass and spring with nothing pushing it, and the spring constant simulator shows where the k in the box comes from.

The last claim is the awkward one. The frequency at which the response is largest is not the natural frequency: it is f0 sqrt(1 - 2ζ2), always a little lower, and it slides further down as the damping grows — 1.591 Hz at a damping ratio of 0.01000, 1.588 Hz at 0.05000, 1.125 Hz at 0.5000. Past 1/sqrt(2), about 0.7071, there is no peak at all. What does stay exactly at the natural frequency, for every damping ratio and with no approximation anywhere in it, is the 90.00 degrees phase lag. That is the honest signature of resonance, and it is the reading to trust.

Where the resonance frequency calculator breaks down

The arithmetic here is exact to the last digit a double can hold. What still goes wrong is physical: an object that has many frequencies rather than one, a spring that is not linear, a loss that is not proportional to speed, and a question about the build-up put to a calculation that only describes the steady state.

Real objects have many modes, and this page describes exactly one

A mass on a spring is the simplest oscillator there is: one coordinate, one frequency. A beam, a panel, a shaft or a casing has a whole series of them, each with its own shape and its own damping, and driving such an object sweeps through one after another. Use this page for the mode you can isolate — one lump of mass on one obvious flexibility — and treat anything more complicated as needing a measurement rather than a formula.

The spring has to be linear, and real springs are not for ever

f0 = (1/2pi) sqrt(k/m) assumes the restoring force is strictly proportional to displacement. A coil spring compressed until its turns touch, a rubber mount squashed flat, or a beam bent far enough to stretch its fibres all stiffen as they deflect, and a stiffening spring has a natural frequency that rises with amplitude. At resonance the amplitude is exactly where it gets large, so this is the assumption most likely to fail in the very state the page is about.

Viscous damping is a model, not a measurement

The damping coefficient box assumes a resisting force proportional to speed, which is what a dashpot or a thin oil film gives. Dry friction, internal losses in a metal and acoustic radiation obey different laws, and a single c fitted to one amplitude will not predict another. Where you have a measured Q instead, work backwards from it: the damping ratio is 1 / (2Q), and c follows as 2ζ sqrt(mk).

Nothing here says how long the build-up takes

Every reading on this page is the steady state, the swing the system settles into after being driven for a while. A lightly damped system takes many cycles to get there — which is exactly why a brief pass through a resonance is survivable when sitting on it is not — but how many cycles, and what the motion looks like on the way, is a separate calculation and this page makes none of it. No figure for a ring-up or ring-down time appears anywhere here.

The magnification is a multiplier, so it needs your own static deflection

There is deliberately no force box and no amplitude box: the whole point is that a small force gives a large response, and a force slider would let you get a big number the boring way. To turn a magnification into a distance, work out the deflection your own steady push gives — the force divided by k — and multiply. A 2 N push on the default 200 N/m spring bends it 10 mm held steady, so a magnification of 10.00 is a 100 mm swing.

The half-power width is an approximation and is labelled as one

The working prints the half-power width as about f0 / Q, and the hedge is not politeness. Against the exact width that approximation runs low by about 0.02 per cent at a damping ratio of 0.01, about 0.08 per cent at 0.02 and about 0.5 per cent at 0.05. For a sharp resonance that is far better than anything you could measure; once the damping is heavy the quantity stops meaning anything at all, because there is no peak to be half the power of.

A peak that formally exists is not a peak you can see

Load the just inside the threshold preset: the peak reading says 0.2251 Hz, which looks like a real answer. Its magnification, though, is 1.000 to four figures — identical to the static response — so there is nothing there to find. By the time the damping ratio reaches 0.70 the peak is already gone in any measurable sense; at 1/sqrt(2) it formally disappears and the reading becomes the word none.

The textbook bridge story is not forced resonance

The 1940 collapse of the Tacoma Narrows bridge is the example every introduction reaches for, and it is the wrong one. The failure is attributed to aeroelastic flutter — a self-exciting oscillation driven by the structure's own motion, feeding on a steady wind rather than on a periodic push at a natural frequency. Billah and Scanlan made the correction in the American Journal of Physics in 1991, and it has been in the literature ever since. Nothing on this page models flutter, and no wind speed, date, frequency or dimension for that bridge is quoted here.

A refusal is an answer, not a fault

A zero or negative mass, a zero or negative spring constant, a zero or negative damping coefficient, a zero or negative drive frequency and a zero or negative natural frequency in the two back-solve directions are all declined in words rather than answered with an infinity or a NaN. The zero damping case is the one worth understanding: it would make Q and the magnification at resonance genuinely infinite, and no real system has it.

The calculator measures nothing — you supply the numbers

Two quantities and two auxiliaries go in, one relation is rearranged, and the answer describes the oscillator you described. The preset names are shorthand for the five figures behind them and are not claims about any particular machine, mounting or installation.

Where the resonance frequency calculator is actually used

Checking whether a machine will sit on its own resonance

This is the forward question and the reason the page opens where it does: a mass you know on a mounting whose stiffness you know, and the frequency to keep the driving machinery away from. Enter the drive frequency you expect and read the magnification. At the default spring and mass a drive at 1.592 Hz is magnified 10.00 times, while the same oscillator driven at 3.183 Hz is magnified only 0.3326 — less than if you pushed and held.

Retuning a mounting by changing the mass

Point Solve for at the mass and the question inverts: a spring you cannot change and a frequency you want to move to. Dropping the default oscillator from 1.592 Hz to 1.000 Hz on the same spring takes 5.066 kg instead of 2 kg — 3.066 kg of added ballast — because the mass goes as one over the square of the frequency. Adding mass is usually cheaper than fighting the amplitude.

Choosing a mounting stiffness for a mass you already have

Point Solve for at the spring constant and you are designing rather than checking. A 0.5066 kg mass that must sit at 1.500 Hz needs a 45 N/m spring; a 0.25 kg mass at 4.000 Hz needs 157.9 N/m. Remember that the stiffness goes as the square of the frequency, so this is the direction in which a sloppy frequency does the most damage.

Working out how much damping you actually need

Load the five damping presets from the lightest damper to the heaviest and watch one reading: 50.00, 10.00, 1.000, 0.7143, 0.7042. That is the whole design trade in five clicks, and nothing but the damper moved between them. It is also where the law of diminishing returns shows up: going from 0.4 to 2 N s/m removes a factor of 5, while going from 20 to 28.4 N s/m removes almost nothing, because by then there is no resonance left to suppress.

Reading a measured resonance backwards

If you have shaken something and found where it responded most, you have the peak rather than the natural frequency, and the two differ by a factor of sqrt(1 - 2ζ2). For a sharp resonance the gap is negligible — 1.591 Hz against 1.592 Hz at a damping ratio of 0.01000. For a heavily damped one it is not: at 0.5000 the peak sits at 1.125 Hz while the natural frequency is still 1.592 Hz, a gap of nearly 30 per cent. The phase lag is the better measurement, because it crosses 90.00 degrees at the natural frequency whatever the damping.

Recognising the same mathematics somewhere else

Every resonance in physics has this shape. A swing pushed in time, a wine glass answering one note, a radio tuned to one station, a nucleus flipped by one radio frequency in an MRI scanner — all of them are a system with a frequency of its own being driven at it, and all of them have a Q. What differs is what plays the part of the mass and the spring. For a pendulum it is the length and gravity, which is the pendulum period calculator; for a stretched string it is a whole ladder of modes at once, drawn by the standing waves simulator.

Where to go next

For what resonance is, how the energy builds up cycle by cycle, the worked problems and the misconceptions, read the resonance in physics guide, which this tool is the arithmetic half of. The simple harmonic motion guide is what the mass is doing once it is moving, the standing waves and harmonics guide is the case where one object has a whole ladder of resonances instead of one, and the amplitude guide is where the height of a vibration belongs as a subject.

Two more posts pick up what this page leaves out. The pendulum period formula guide is the same shape of formula with a length and gravity in place of a spring and a mass, and the frequency formula guide is the period-to-frequency conversion for a reader who arrives with a period.

Four tools sit next to this one. The simple harmonic motion calculator handles the angular frequency, the period and the peak speed; the spring constant calculator supplies the k this page asks for; the Hooke's law calculator is the force-extension relation behind it; and the pendulum period calculator is the same algebra on a different system.

The simple harmonic motion simulator, the spring constant simulator and the standing waves simulator are the neighbouring simulations. The full physics lab library and calculator index are open too.